Abstract

By M-matrix theory, inequality techniques, and Lyapunov functional method, certain sufficient conditions are obtained to ensure the existence, uniqueness, and global exponential stability of periodic solution for a new type of high-order BAM neural networks with continuously distributed delays and impulses. These novel conditions extend and improve some previously known results in the literature. Finally, an illustrative example and its numerical simulation are given to show the feasibility and correctness of the derived criteria.

1. Introduction

As is well known, during the hardware implementation of neural networks, time delays are inevitable due to finite switching speeds of the amplifiers and communication time, which may bring about complex influence on the system such as oscillation and instability [1, 2]. On the other hand, impulsive effects wildly exist in many realistic networks [3, 4], which may be caused by witching phenomenon, sudden changes, or other unexpected noise. Therefore, it is more appropriate to consider delay and impulsive effects when modeling neural networks, and many researches on various kinds of neural networks with delays, impulses, or both of them have been available [5–12]. (See Figures 1(a), 1(b), and 1(c)).

Figure 1: It is confirmed that the system (51) has a unique -periodic solution. Here and denote two pairs of solution of system (51) with different initial conditions and for and .

Bidirectional associative memory (BAM) neural networks, as an extension of the unidirectional autoassociator of Hofield neural network [13], was firstly introduced by Kosko [14]. Due to its wide application in pattern recognition, associative memory, image, and signal processing, BAM neural networks with delays and impulses have been extensively studied in the past few decades [15–22]. In addition, it is worth noting that high-order neural networks structures have advantages of stronger storage capacity, faster convergence rate, and higher fault tolerance, and these merits have been successfully used in pattern recognition [23]. Thus, it is important to investigate BAM neural networks with high-order terms, which is called high-order BAM neural networks.

In this paper, we will consider a new type of high-order BAM neural networks with continuously distributed delays and impulses, which can be described by the following integrodifferential equations:
where and are the impulses at moments and is a strictly increasing sequence such that . And and are the activations of the th neuron and the th neuron, respectively; and denote the passive decay rates; are the first- and second-order connection weights of the neural networks, respectively; and are the external inputs.

Clearly, system (1) is a more general form of BAM neural networks, which has been widely applied in areas of science and engineering [24], such as neurobiology, image classification, and image recognition. In recent years, studies of such kind of neural networks with delays and impulses have received considerable interest, and some results have been reported in [25–30]. In particular, authors in [25–28] have discussed the stability of equilibrium point for a kind of impulsive high-order BAM neural networks with discrete delays by different methods, such as linear matrix inequality (LMI), Razumikhin technique. Subsequently, Huo et al. [29] and Yang [30] studied the existence of periodic solution and its exponential stability for an impulsive high-order BAM neural network with discrete delays by using the theory of coincidence degree and Lyapunov functional method. However, to the best of our knowledge, there are few results on the existence, uniqueness, and global exponential stability of periodic solution for system (1) with continuously distributed delays.

The main propose of this paper is to study the periodicity of system (1) with distributed delays and general impulsive effects. It should be noticed that some new criteria on the existence and uniqueness of periodic solution for system (1) are established by combining the general (see Notations) and analytical techniques, which is different from the conventional continuation theorem of coincidence degree theory used in [29, 30]. In addition, it is worth mentioning that the impulsive part in this paper is not necessarily bounded and linear, which makes its applications more extensive.

The rest of this paper is organized as follows. In Section 2, some assumptions, definitions, and important lemmas are given. In Section 3, the main results and some remarks are presented. In Section 4, an example and its numerical simulation are provided. Finally, some conclusions are summarized in Section 5.

2. Preliminaries

Notations. Throughout this paper, and denote the set of real numbers and -dimensional vector space, respectively. The symbol denotes the transpose of a vector or a matrix. Take with integer . Clearly, are special cases of with , respectively, which are used to investigate the dynamics of various kinds of neural networks in [6–8, 10, 12, 15–17, 19, 21, 22, 25–31]. Denote is continuous for all but at most countable points and at these points , and exist, and define the norm by
where , and then is a Banach space with topology of the uniform convergence. In addition, system (1) is supplemented with initial values
As usual, we have the following assumptions for system (1).

(S1) Functions ,and are -periodic and bounded on such that for .

(S2) The activation functions , and are bounded and Lipschitz continuous on ; that is, there exist positive numbers and such that
for and .

(S3) The delay kernel functions , and are piecewise continuous functions from to and satisfy ,, and for , , where satisfy

in which denotes some positive constant number. For more information on these delay kernels, one can refer to [5, 9, 15, 18, 22].

(S4) is an -matrix, where ,

Definition 1. A function , ,…, is said to be the solution of system (1) with initial condition if the following two conditions are satisfied.(1) is piecewise continuous with first kind discontinuity at the points ,. Moreover, is left continuous at each of the discontinuity points. (2) satisfies system (1) for and for .

Definition 2. The periodic solution of system (1) is said to be globally exponentially stable, if there exist constants and such that any other solution of system (1) satisfies

Lemma 3 (see [32]). Let , where is a set of matrices with nonpositive off-diagonal elements. is an -matrix if and only if there exists a positive vector such that or .

Lemma 4. Assume that assumptions (S3) and (S4) hold; then there exist positive constants and such that
for .

Proof. Construct the aided functions as follows:
for . Using Lemma 3 and assumptions((S3)-(S4)), it is easy to deduce that (8) hold by similar proof in [7, 15–18]. For concise, it is omitted here.

Lemma 5. Let the integer ; then the inequality holds as follows:
for all .

Proof. Obviously, the inequality (10) with is trivial. When , consider the aided function . It is claimed that is convex since , for . Let ; by Jensen's inequality, we have
which implies that the inequality (10) holds. This completes the proof.

3. Main Results

Firstly, let , and ,,…, be any two solutions of system (1) through , respectively; then we have the following useful lemma.

Lemma 6. Under assumptions ((S1)–(S4)), if the following two conditions hold: (S5) are Lipschitz continuous on ; that is, there exist positive constants such that for and ;(S6)there exists such that , where , ,, and the scalar is estimated by (8).Then, the following inequality holds:
where the constants and are to be determined later.

Proof. To be convenient, let
It follows from ((S1)–(S3)) that
for , . Similarly, we have
for , . Also,
Now define
By using Young inequality , where and , it follows from (14)-(15) that
for , . Similarly, we have
for , . Also
Consider the candidate Lyapunov-Krasovskii functional as follows:
where
When , calculating the upper right Dini derivative of along the solutions of system (1), we get
Similarly, we have
Therefore, by Lemma 4, we obtain that
When , we have
Now, we claim that
In fact, for , noticing that and (25), we have
On the other hand, from (26), we have
Combining (28) and (29), we obtain
which implies that (27) holds for . Assume that (27) holds for , that is,
Then, for , from (25), we have
On the other hand, from (26), we have
From (31)–(33), we obtain
This shows that (27) holds for . Hence, by mathematical induction, (27) holds for all . Combining (25) and (27), we obtain
for all . Noticing that in , we have
for all . On the other hand, it follows from (21) that
where
Together with (36)-(37), we have
for all . Let , and then we have
This completes the proof.

In the following, we will study the existence, uniqueness, and global exponential stability of periodic solution of system (1) by exploiting Lemmas 5 and 6.

Theorem 7. Assume that assumptions (S1)–(S6) hold, then system (1) has a unique -periodic solution, which is globally exponentially stable.

Proof. Firstly, we prove the existence of periodic solution of system (1). To this end, let , be an arbitrary solution of system (1) through , where . Define , where . We can know that and is also a solution of system (1) through . By virtue of Lemma 6, we have
for . So, we have
for . It follows from Lemma 5 that
Noticing that for ,
It follows from (43)-(44) that
which implies that exists. Similar to (44) and (45), we obtain that exists. Let
where for . Then is an -periodic solution for system (1).Secondly, we prove the uniqueness of periodic solution of system (1). Assume that is another -periodic solution of system (1) through , where . By a minor modification of the proof of (43), we have
Taking , we have
which implies that system (1) has a unique -periodic solution.Finally, since is a unique -periodic solution of system (1), let be any other solution of system (1) through . From Lemma 6, we obtained that
where and are the same as defined in Lemma 6. It follows from Definition 2 that the -periodic solution is globally exponentially stable. Up to now, we conclude that system (1) has a unique -periodic solution , which is globally exponentially stable. This completes the proof.

Remark 8. In assumption (S5), we only assume that the impulsive operators and are Lipschitz continuous, which remove the usual assumptions that the boundedness and linearity of the impulsive operators are required in [18, 19, 21, 29–31]. Thus, our results have wider adaptive range. Particularly, if we take the linear operators and as considered in [18, 19, 21, 29–31], that is,(S7)then we have . So we can choose and to satisfy assumption (S6). In this case, we have the following interesting corollary.

Corollary 9. Assume that assumptions ((S1)–(S4)) and (S7) hold; then system (1) has a unique -periodic solution, which is globally exponentially stable.

Remark 10. Note that when in assumption (S6), which implies that there are no impulsive effects on system (1). Correspondingly, we call system (1) an impulse-free. In this case, we have the following corollary.

Corollary 11. Assume that assumptions ((S1)–(S4)) hold; then the impulse-free system (1) has a unique -periodic solution, which is globally exponentially stable.

Remark 12. Clearly, based on the general and Lemma 5, a general criterion ensuring the existence of periodic solution and its global exponential stability of system (1) with and without impulses has been established. Compared with results in [6, 7, 15–17], it is easy to see that our results are extended and improved because their results can be viewed as the special case of in assumption (S4). In addition, since the nonnetwork parameter is introduced in the condition (S4), it can allow much broader applications for designing the circuit of a convergent impulsive network.

Remark 13. In assumption (S3), if the kernel is a delta function of the form:
where and , then system (1) with continuously distributed delays reduces to the model with discrete delays in [29]. According to Lemma 3, we know that the condition (H5) of Theorem in [29] implies that with is an -matrix but not vice versa. Thus, our results are new and complementary to their results.

4. An Example

In this section, an example and its numerical simulation are given to illustrate the correctness of the obtained theoretical results.

An Example. Consider the following high-order BAM neural networks with infinite distributed delays and impulses:
where . By simple calculation, we obtain that and
If the integer , then is an -matrix. Thus, assumptions ((S1)–(S4)) are satisfied for system (51). For the impulsive part, the following two cases are considered.

Case 1. When , by Corollary 11, we conclude that the impulse-free system (51) has a unique -periodic solution, which is globally exponentially stable.

Case 2. When the impulsive parts are taken as the nonlinear operators such that and , that is,
which satisfy the assumption (S5) with Lipschitz constants . So we can choose and to satisfy the assumption (S6). According to Theorem 7, we conclude that the impulsive system (51) has a unique -periodic solution, which is globally exponentially stable. However, in this case, results in [18, 19, 21, 29–31] are ineffective because the function is nonlinear. Moreover, the condition (H5) of Theorem in [29] cannot be applied to system (51), since .

5. Conclusions

In this paper, we have studied the existence, uniqueness, and global exponential stability of periodic solution for a kind of high-order BAM neural networks with continuously distributed delays and general impulses. It should be noted that some extended and improved criteria have been derived by exploiting the general , Lemma 5, and the Lyapunov functional method. In addition, these criteria are in terms of -matrix, which can be easily checked by many equivalent conditions listed in [32]. Finally, an example and its numerical simulation are given to show the feasibility and correctness of the obtained results.

Acknowledgments

The authors appreciate the anonymous reviewer for her/his insightful comments and valuable suggestions, which have led to a great improvement in this paper. This research is supported by the Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), the Research Fund of Kunming University (XJ11L019, XJL11008), and the Fundamental Research Funds for the Central Universities (ZYGX2010J106).